How do you find all solutions of the equation #sin(x+pi/4)-sin(x-pi/4)=1# in the interval #[0,2pi)#?

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Answer 1 · 2017-06-15T14:18:50

#"The Soln. Set.="{2kpi+-pi/4 | k in ZZ.}.#

The Solns. in #[0,2pi), are, therefore, pi/4, and, 7pi/4.#

#sin(x+pi/4)-sin(x-pi/4)=1.#

#rArr 2cos[1/2{(x+pi/4)+(x-pi/4)}]sin[1/2{(x+pi/4)-(x-pi/4)}]=1.#

#rArr 2cosxsin(pi/4)=1.#

#rArr2*1/sqrt2*cosx=1.#

#rArr cosx=1/sqrt2=cos(pi/4).#

Knowing that, #costheta=cosalpha rArr theta=2kpi+-alpha, k in ZZ,#

#"The Soln. Set.="{2kpi+-pi/4 | k in ZZ.}.#

In Particular, the Solns. in #[0,2pi), are, therefore, pi/4, and, 7pi/4.#